Appendix: Final form of the algebraic mass ux model The pertinent facts concerning the computational implementation of the mass ux representation are presented here. The general form of the model, valid for a three-dimensional ow in an inertial system without body forces, is <v i >=[ 0 ij + 1 V i ; j + 2 2 V i ; k V k ; j ]fv j v p g <>; p <> 1 where the viscosity coecients are functions of the invariants of, A = 1 + rv, and are given by 0 =(1 I A +II A )=III A, 1 =(2 I A )=III A, and 2 =1=III A. The invariants for the tensor are given by I A = <A> II A = 1=2(< A> 2 <A 2 >) III A = 1=6(< A> 3 3< A >< A 2 > +2 <A 3 >) for which <> indicates the trace of the enclosed matrix. =(M t k=")=(1 + M t (P=" The time scale is dened as 1)) The various traces are straightforward to compute using their denition. Their signicance can be understood when they are recast in terms of the mean dilatation, rotation and strain: <A>= 3+D <A 2 >= 3+2D + 2 [<S 2 >+<W 2 >] <A 3 >= 3+3D +3 2 [<S 2 >+<W 2 >]+ 3 [<S 3 >+2 <SW 2 >+<W 3 >] D = S jj is the mean dilatation and the strain and rotation tensors are dened: S ij = 1=2[V i ; j +V j ; i ] and W ij =1=2[V i ; j V j ; i ]. The term D can be thought of as a ratio of uctuating to mean dilatation time scales. D < 1. It is understood to be order one or smaller, 18
References Blaisdell, G.A. (1991). Numerical simulations of compressible turbulence. PhD thesis, MAE, Stanford University, Stanford CA. Dinavahi, S.P.G., C. D. Pruett (1993). Analysis of direct numerical simulation data of a Mach 4.5 transitional boundary layer ow. ASME Fluids Engineering Conference, June 21-24, Washington DC. Dinavahi, S.P.G., J.R. Ristorcelli, G. Erlebacher (1993). Some turbulence statistics relevant to compressible turbulence modeling. To appear in the Proceedings of the ICASE/LaRC Workshop on Transition, Turbulence and Combustion. Gatski. T. (1993). personal communication. Lee, S. (1992). Interaction of isotropic turbulence with a shock wave. PhD thesis, MAE, Stanford University, Stanford CA. Ristorcelli, J.R. (1993). Carrying the mass ux terms exactly in the rst and second moment equations of compressible turbulence. ICASE Report No. 93-87, NASA CR-191568, submitted Phys Fluids. Ristorcelli, J.R., S.P.G. Dinavahi (1993). Testing a model for the turbulent mass ux. To appear in the Proceedings of the ICASE/LaRC Workshop on Transition, Turbulence and Combustion. Ristorcelli, J.R., S.P.G. Dinavahi, G. Erlebacher (1993). Turbulence statistics relevant to compressible turbulence modeling. To be submitted to Phys Fluids. Rubesin, M.W. (1990). Extra compressibility terms for Favre averaged two equation models of inhomogeneous turbulent ows. NASA Contractor Report 177556. Taulbee, D., J. VanOsdol (1993). Modeling turbulent compressible ows: the mass uctuating velocity and the squared density. AIAA 91-0524, 29 st AIAA meeting, Reno, NV. Zeman, O. (1993). A new model for super/hypersonic turbulent boundary layers. AIAA 93-0897, 31st AIAA meeting, Reno, NV. Zeman, O., Coleman, G.N. (1991). Compressible turbulence subjected to shear and rapid compression. Proceedings of the Eight Symposium on Turbulent Shear Flows, Springer- Verlag. 17
turbulence elds in which the production terms play a major role. For ows in which the pressure dilatation covariance plays a major role in transferring energy from its kinetic to potential modes it may be necessary to reevaluate the adequacy of the linear relaxation model. The form of the mass ux model presented does not include eects associated with rotation or body forces. Both of these eects can be easily incorporated as they do not require any additional modeling; it is simply a matter of retaining the extra terms in the algebraic truncation of the evolution equations. There is an exception; at rapid rotation rates the neglected pressure covariance becomes important and the truncation of the evolution equation used to obtain the model is no longer valid. The model is realizable for most simple ows though a general proof of its realizability has not been found. These realizability aspects, and the fact that the destabilizing properties associated with isotropic eddy-viscosity models do not appear in this mass ux model, are expected to make it computationally robust. In the moment evolution equations for a compressible turbulence the mass uxes appear in several places. In the mean momentum and energy equations the mass ux appears in ve dierent locations, Ristorcelli (1993), and modeling U i ' V i ignoring the contribution of the mass ux has been shown to be inadequate. In the Reynolds stress equations the mass ux determines the relative importance of the production by the mean ow acceleration, it contributes to the pressure uxes and the viscous uxes. It is clear, given the number of times it occurs in the moment evolution equations, that an accurate model for the mass ux is necessary for complex compressible turbulent ows of aerodynamic interest. This is to assess the magnitude of the mass ux in various ows and to include it in a computational model when it is important. There are classes of compressible ows in which the contribution from the mass ux are expected to be small and its inclusion in a computational model is unnecessary. It is expected that the mass ux will not make much of a contribution to usual unidirectional shear ows such as the at plate boundary layer and diverse free shear layers, unless there are large density gradients. The mass ux terms are expected to be important in more complex ows: these include ows in which there are mean density gradients due to large Mach number or combustion, separation or reattachment (inection points), cold wall boundary conditions, mean dilatation, shocks, adverse pressure gradients, or strong streamwise accelerations such as those occurring in ramp type ows. 16
appearance of the mean velocity gradients in the tensor eddy-viscosity reects their presence in the production terms in the evolution equation for <v i >. The model predicts countergradient transfer and shows that mean density gradients in one direction can produce a mass ux in a dierent direction. It form, valid for a general three-dimensional ow, is <v i >= <><v i >= [ 0 ij + 1 V i ; j + 2 2 V i ; k V k ; j ]fv j v p g <>; p where =(M t k=")=(1 + M t (P=" 1)). The viscosity coecients, 0 ; 1 ; 2, are known functions of the mean velocity gradient, given in terms of the invariants of the tensor A = 1 + rv. They are not adjustable "tuning" coecients. The derivation of the expression for <v i >has involved a minimum number of assumptions regarding the physics of compressible turbulence. It is useful, however, to keep in mind some of the approximations to account for possible discrepancies and to anticipate the classes of ows in which the present form of the model may be inadequate. The assumptions used are: 1) The derivation of an O (< > 1=2 =<>) set of evolution equations for the <v i > showed that the unclosed terms involving correlations with the uctuating pressure and stress are higher order eects and can therefore be neglected. In the evolution equations there is only one unclosed term, the uctuating <v i v k ; k >covariance. 2) The form of turbulent diusion terms appearing in the <v i >equation, are found to scale with the density intensity, <> 1=2 =<>, for arbitrary inhomogeneity and can therefore be neglected. This is consistent with the truncation of the equation as (< v i v p ><> 1 ); p = (fv i v p g <v i v p >); p is an O (< > 1=2 =<>) quantity. The dierence between fv i v p g and <v i v p >has been seen to be small in the wall bounded ow at Ma =4:5 of Dinavahi and Pruett (1993), as seen in Ristorcelli et al. (1993). 3) The structural equilibrium assumption, D=Dt [< v i >< v j >=fv p v p g] = 0, is used to produce an algebraic expression for the mass ux equation. This allows the material derivative to be expressed in terms the production and dissipation of the turbulence energy. For more rapidly varying ows in which the structural equilibrium is not expected to yield results of adequate accuracy it is possible to carry the full dierential equation for the mass ux. Near solid boundaries, were the mass ux is most important, a structural equilibrium is expected to be achieved rapidly and the algebraic form is adequate. It is this fact, coupled with the density intensity truncation of the evolution equation, that enables the mass ux expression to be used all the way to the wall without any ad hoc wall function corrections. 4) The algebraic truncation of the evolution equation for <v i >involves one unclosed term,< v i v k ; k >. It has been assumed that it can be represented as a linear relaxation term, <v i v k ; k >= <v i >= d where d = M t k=". This model for the covariance with the uctuating dilatation is expected to be adequate for most quasi-equilibrium quasi-homogeneous 15
Case 5: Arbitrary two-dimensional mean velocity gradients For an arbitrarily complex two-dimensional ow, such as the developing wall bounded turbulent boundary layer with separation, V i ; j =[V 1 ; 1 ;V 1 ; 2 ;0]; [V 2 ; 1 ;V 2 ; 2 ;0] is a suitable representation for the velocity gradient eld. The viscosity coecients are given by 0 =1, 1 = (1 + D)=III A, 2 =1=III A where D = V 1 ; 1 +V 2 ; 2. The mass uxes are given by <v 1 >= III A [(1 + V 2 ; 2 )fv 1 v p g fv 2 v p gv 1 ; 2 ] <>; p <> 1 <v 2 >= III A [(1 + V 1 ; 1 )fv 2 v p g fv 1 v p gv 2 ; 1 ] <>; p <> 1 where III A =1+D + 2 (V 1 ; 1 V 2 ; 2 V 1 ; 2 V 2 ; 1 ). Case 6: Arbitrary three-dimensional strain with simple shear In a general three-dimensional ow the expressions for the invariants are somewhat more complicated. The simplest case, a simple shear with arbitrary three dilatation, is chosen. The velocity gradients are represented by V i ; j =[V 1 ; 1 ;V 1 ; 2 ;0]; [0;V 2 ; 2 ;0]; [0; 0;V 3 ; 3 ]. The square of the velocity gradientisgiven by V i ; k V k ; j =[(V 1 ; 1 ) 2 ;V 1 ; 2 (V 1 ; 1 +V 2 ; 2 );0]; [0; (V 2 ; 2 ) 2 ; 0][0; 0; (V 3 ; 3 ) 2 ]. The invariants of A are I A =3+D, II A =3+2D + 2 (V 1 ; 1 V 2 ; 2 +V 2 ; 2 V 3 ; 3 +V 3 ; 3 V 1 ; 1 ), and III A =(1+V 1 ; 1 )(1 + V 2 ; 2 )(1 + V 3 ; 3 ). Here, as usual, D = V j ; j is the mean dilatation. The viscosity coecients are a little more complicated - the three-dimensionality of the ow now aects the zeroth-order viscosity coecient. In the two-dimensional ows 0 =1; here 0 =(1+D + 2 d(v 1 ; 1 V 2 ; 2 +V 2 ; 2 V 3 ; 3 +V 3 ; 3 V 1 ; 1 ))=III A. The higher order viscosity coecients are given by 1 = (1 + D)III A, 2 =1=III A and the uxes are written as <v 1 >= III A [(1 + (V 2 ; 2 +V 3 ; 3 )+ 2 V 2 ; 2 V 3 ; 3 )fv 1 v p g fv 2 v p gv 1 ; 2 (1 + V 3 ; 3 )] <>; p <> 1 <v 2 >= III A [(1 + (V 1 ; 1 +V 3 ; 3 )+ 2 V 1 ; 1 V 3 ; 3 )fv 2 v p g]<>; p <> 1 6. Summary and Conclusions The uctuating Favre velocity mean, <v i >, is the rst-order form of a second-order moment, the mass ux, <v i >= < >< v i >. The mass uxes quantify the dierence between Reynolds statistics and the density-weighted Favre statistics, U i = V i + <v i >and u i = v i <v i >, and can be thought of as measuring the eects of compressibility due to variations in density. The eects of the mean density gradients on the anisotropy of the turbulence are fully parameterized by the mass ux. An algebraic representation for the mass ux has been derived from the transport equation for the Favre uctuation mean using the structural equilibrium assumption. The mass ux is found to be proportional to the mean density gradients with an anisotropic eddyviscosity that depends on both the Reynolds stresses and the mean velocity gradients. The 14
isotropic eddy-viscosity model cannot predict; such a model predicts a zero streamwise mass ux. The predictions of the cross-stream component, <v 2 >, are less successful. This is because there are no large production terms in the <v 2 >expression to mask the inaccuracies of the linear relaxation model assumed for the dilatational correlation in a nonequilibrium "newly formed" turbulence. The present temporal DNS is started from a laminar prole and computed through transition. The data shown in the gures represents a ow approximately three eddy-turnovers past the transition, "x=(ku) ' 3. The turbulence eld is not fully developed, retaining vestiges of the initial conditions; a linear relaxation model for the correlation with the uctuating divergence would not be expected to do well in such, a more or less, transitional ow. The poor agreement in the expression for the cross-stream mass ux cannot be explained by the fact that the data comes from a temporal DNS. The expression for the mass ux model is from its evolution equation which is independent of the mean ow equation calculation. Thus, the problem often seen in comparing Reynolds stress model calculations to temporal DNS simulations, in which there is a forcing term in the equations to compensate for the boundary layer growth, do not appear here. Case 4: Plane strain with mean dilatation For a plane strain with arbitrary non-zero dilatation, V i ; j = V 1 ; 1 i1 j1 +V 2 ; 2 i2 j2, and the viscosity coecients take on the following simple values 0 =1, 1 = (1+(V 1 ; 1 +V 2 ; 2 ))=III A, 2 =1=III A where III A =(1+V 1 ; 1 )(1 + V 2 ; 2 ). The uxes are given by the simple expressions <v 1 >= fv (1+V 1 ; 1 ) 1v p g <>; p <> 1 <v 2 >= fv (1+V 2 ; 2 ) 2v p g <>; p <> 1 : Clearly the model is fully realizable and the destabilization of more rudimentary models, noted by Zeman and Coleman (1991), for a ow with large normal strain is not an issue. Note that in very high strains, say the normal passage through a shock, the dependence on the phenomenological parameter d, absorbed in is lost. Here, again, =(M t k=")=(1 + M t (P=" 1)). In mean mean eld with a large dilatational component, it is more useful to consider a mean velocity gradient described of the form V i ; j = V 1 ; 1 ((1+D) i1 j1 i2 j2 ). The viscosity coecients take on the following values 0 =1, 1 = 1(1 + DV 1 ; 1 )=III A, 2 =1=III A. The streamwise and cross-stream components of the Favre uctuation mean, in a ow with arbitrary density gradient, become <v 1 >= 1+(1+D)V 1 ; 1 fv 1 v p g <>; p <> 1 <v 2 >= 1 V 1 ; 1 fv 2 v p g <>; p <> 1 13
in which the eigenvalue of the Reynolds stress vanishes will also vanish. It is interesting to compare the expression for the mass ux to Zeman's (1993). In an equilibrium turbulence, for which P = ", and in the limit of small velocity gradients the present model simplies, to within a constant of proportionality, to Zeman's model. Comparisons of this model with the DNS data in a wall bounded compressible ow shows that it does not successfully capture the results known from DNS. The neglected terms involving the velocity gradients are essential. After all the terms with the mean velocity gradients represent the production terms, which are typically not negligible, in the mass ux equation. Computations with the neglected velocity gradient capture the near wall behavior very nicely, as will be seen in the next case. Case 3: Simple shear, V i ; j = V 1 ; 2 i1 j2 In problems of engineering interest the turbulence will be anisotropic and there will be nonnegligible gradients in the mean velocity eld and the production terms for the mass ux need to be included in the algebraic expression. For the simple shear, V i ; j = V 1 ; 2 i1 j2,a surprisingly simple expression for the mass ux is possible. The computation is easily carried out by hand using <v p >( ip + V i ; p )=fv i v p g<>; p <> 1. Using the inversion formula the invariants of A are I A =3; II A =3; III A = 1 and the viscosity coecients take on the simple values 0 =1; 1 = gradient is zero, the expression for the mass ux becomes 1; 2 = 1 and, as the square of the mean velocity <v i >=[ ij V 1 ; 2 i1 j2 ]fv j v p g <>; p <> 1 The streamwise and cross-stream components of the Favre uctuation mean, in a ow with only a cross-stream density gradient, become <v 1 >=[fv 1 v 2 g V 1 ; 2 fv 2 v 2 g] <>; 2 <> 1 <v 2 >=fv 2 v 2 g<>; 2 <> 1 : Note that the eects of the production of <v i >by the mean shear, proportional to V 1 ; 2, are included in the expressions for the streamwise mass ux. This is similar to the normal Reynolds stresses in a unidirectional shear: the production mechanism is in the equations of the streamwise component of the energy and therefore it is larger than the spanwise and cross-stream components of the turbulence energy. Computations, shown in Figure 3, with this model are very successful for the streamwise component <v 1 >. The peak in <v 1 >is captured surprisingly well in size and location. This behavior can not be captured without the inclusion of the V 1 ; 2 fv 2 v 2 g term. The small velocity gradient limit expression, case 2 above, which is essentially the algebraic form both Zeman and Cole (1991) and Rubesin (1990) substantially underpredicts the near-wall peak of the mass ux. Note that a streamwise mass ux is engendered by a cross-stream density gradient. This is a behavior that an 12
Case 1: Isotropic turbulence with small velocity gradients, rv << 1 In this case rv << 1, P = 0 and the time scale =(M t k=")=(1 + M t (P=" eddy-viscosity tensor assumes the form fv j v p gm t (k=")2k jp =(1 <v i > Mt 1 M t (k=")2k <>; i <> 1 : 1)). The M t ) and the model is This can be compared to the usual eddy-viscosity model: <v i >=( T =<> 2 Pr t ) < >; j in which T = C f <>k 2 = and thus <v i >(k=")k <>; i <> 1 : The usual eddy-viscosity form misses the dependence on M t which is necessary if the <v i >are to vanish in the absence of compressibility eects. Thus, apart from the M t scaling, a scalar viscosity assumption will work, in the limit of an isotropic turbulence with negligible mean velocity gradients. Note that this form in a boundary layer ow with crossstream density gradient cannot predict a streamwise mass ux. It can only predict a mass ux down the density gradient. In problems of engineering interest there will be countergradient transport, as has been seen in the Ma =4:5 data of Dinavahi et al. (1993), and an eddyviscosity gradient transport hypothesis is inappropriate. These inadequacies have also been noted by Taulbee and VanOsdol (1991). The major shortcoming of the eddy-viscosity assumption is realizability and its impact on computability in compressible closures. In the Reynolds stress equations for arbitrary mean ow accelerations, a gradient transport assumption for <v i >can cause the acceleration production mechanism to destabilize the computations. For example if fv v g, in the Reynolds stress equations above, vanishes, <v >must also vanish in order to keep that eigenvalue of the Reynolds stress from going negative, as a nite <v i >in <v i >< > D=Dt V j will cause negative energies for arbitrary mean acceleration. This cannot be accomplished with the eddy-viscosity form of the model. Gatski (1993) has used a scalar eddy-viscosity representation and found it to be computationally destabilizing. Zeman and Coleman (1991) have also pointed out that inadequate representations of the mass ux can destabilize computations in ows when the acceleration terms is important. This occurs, for example, in the passage through a shock orinows in which the mean strain or mean dilatation are important. Case 2: Anisotropic turbulence with small mean velocity gradients, rv << 1 In the case rv << 1 and when the turbulence is anisotropic the expression for the mass ux becomes where =(M t k=")=(1+m t (P=" <v i >=fv i v p g<>; p <> 1 1)). Note that this expression for the mass ux allows for countergradient transfer and is realizable: the mass ux in the direction of the principal axis 11
signicant features: 1) the mass ux in one direction, as might be expected from continuity considerations, is inuenced by the mass ux in another direction and 2) the contraction of the density gradient on the Reynolds stress allows countergradient transfer. In simple cases this set of equations is easily solved by hand. Performing the general inversion the model can be written in symbolic form as <v i >=T ij fv j v p g <>; p <> 1 where T ij =( ij + V i ; j ) 1. This is an anisotropic eddy-viscosity model in which the eddyviscosity tensor jp = T ji fv i v p g is a function of the Reynolds stresses and the mean ow gradients. Though this form suggests the structure of the model it is not in a form most suitable for computation. Recourse to the Cayley-Hamilton theorem allows T ij to be written in terms of the invariants and the rst and second powers of the matrix: III A A 1 = A 2 I A A + II A 1 Substituting A = 1 + rv produces an expression for the inverse III A (1 + rv) 1 = (1 I A +II A )1 +(2 I A )rv+ 2 (rv) 2 and the nal model can be written, in ascending powers of ratios of time scales, as <v i >=[ 0 ij + 1 V i ; j + 2 2 V i ; k V k ; j ]fv j v p g <>; p <> 1 : The nondimensional "viscosity" coecients, 0 ; 1 ; 2 are known in terms of the mean deformation; they are not phenomenological parameters that require calibration to experiments which then limit the application of the model to ows not too dierent from that for which the model has been calibrated. Only one phenomenological assumption - to obtain the relaxation model for the correlation with the uctuating dilatation - has been made. The invariants and the viscosity coecients are given in the Appendix that summarizes the nal form of the model. 5. Discussion and implementation of the mass ux model in simple ows Formidable as the algebraic expression for the mass ux may appears there are some simple expressions for the <v i >possible. Though the representation is valid for arbitrary three-dimensional ows several cases with two-dimensional mean elds are investigated in order to understand the eects of dierent mean deformations. One three dimensional eld is considered in order to anticipate the eects the three-dimensionality of the ow might have on the mass ux expressions. 10
form of the evolution equation for Favre uctuation mean is now: <v i >(P ")=k = <v p >V i ; p +fv i v p g<>; p <> 1 +<v i v p ; p >: The body force terms and the Coriolis terms have not been carried, however the analysis can be carried quite easily with them as they do not constitute unknown terms that require closure. It remains to close the last term on the right hand side. It is possible, in situations with large density and velocity gradients, to neglect the correlation with the uctuating divergence. This is equivalent to the assumption that the mean ow gradients of density and velocity are large and set the balance to lowest order. It can, however, be shown that the correlation with the uctuating divergence scales with mean ow gradients and is therefore not negligible in a general ow. Moreover there are times when the dierence between the mean production terms is small which means that the contribution from <v i v p ; p >will be important. The correlation with the divergence will be represented by a linear relaxation model. This linear relaxation model is chosen on the grounds that <v i v p ; p >and <v i >have the same tensorial properties. Both belong to the same symmetry groups, satisfying the same reectional and rotational properties, vanishing in isotropic turbulence and in an equilibrium homogeneous turbulence. From a computational point of view, a linear relaxation form is desirable as it avoids the possibility of a singularity in the inversion of the velocity gradient during a computation and is consistent with realizability. A linear relaxation with time scale d <v i v p ; p >= <v i >= d is chosen. Zeman and Coleman (1991) have also used a linear relaxation with acoustic time scale for this correlation. The time scale, d, in the model for <v i v p ; p >= <v i >= d may be thought of as a dilatational time scale. The time scale, d, is the only phenomenological parameter assumed to obtained the present model for the mass ux. Computations using the acoustic time scale, d = M t k=", to represent the dilatational time scale have been successful. The present model will use this approximation for the dilatational time scale. There are other possibilities though at this time, given the success of the present model, there is little motivation for further investigation. Substituting for the unknown correlation with the uctuating divergence in the algebraic truncation of the evolution equation for <v i >produces <v p >( ip + V i ; p )=fv i v p g<>; p <> 1 : where =(M t k=")=(1 + M t (P=" 1)) The model is now a set of three coupled linear algebraic equations of the form A ip <v p >=b i. Inspection of the equations reveals two 9
an evolution equation independent of complicating correlations with the pressure and viscous surface forces. Note that the fact that fv i v j g <v i v j >=<v i v j >< > 1 and that < v i v p >=< >fv i v p ghas been used. The inhomogeneous diusion term [fv i v j g <v i v j >] is an O ( p ) term, as can be seen by the data presented in Dinavahi et al. (1993), and can in general be neglected. In a homogeneous turbulence it is zero, of course. This very simple equation for <v i >results from the fact that, in the Favre setting, surface forces are carried using the Reynolds decomposition while volume forces appear naturally in the Favre variables. The rst-moment of the uctuating surface forces (pressure, viscosity) appearing in the equation for <v i >are zero and no complicating models for these terms are required. This combined with the peculiarity of the uctuating Favre mean allows <v i >= < >< v i > and leads one to work with the rst-moment form <v i >of the second-moment <v i >.Thus a simple evolution equation for the mass ux that highlights the zeroeth order eects associated with the volumetric compressibility while relegating the higher order eects of the surface forces to a higher order equation in the expansion is obtained. 4. An algebraic expression for the Favre-velocity perturbation To obtain the mass ux, <v i >= <><v i >, an equation for the Favre uctuation mean, <v i >, with only one unknown term, the correlation with the uctuating divergence <v i v p ; p >, has been derived. The evolution equation obtained for the Favre uctuation mean is simple enough to carry in turbulence simulations. However it is still simpler and less expensive to carry an algebraic expression. This is now derived. A direct algebraic truncation of the evolution equation will describe the xed points of the <v i >. An algebraic truncation following the procedure used in algebraic stress models will give the xed points of <v i >=fv p v p g 1=2. This is done by assuming a structural equilibrium of the form D=Dt [< v i >=fv p v p g 1=2 ] = 0 allowing the convective derivatives, D=Dt < v i > to be expressed in terms of the right hand side of the evolution equation for the turbulence energy: D Dt <v i>= <v i> D fv p v p gdt fv qv q g =< v i >(P which allows the evolution of the <v i >to reect the changes in the energy of the local turbulence eld. Here P, ", are the production and the dissipation in the turbulent kinetic energy equation where k =1=2fv p v p gis the specic kinetic energy. In the near wall region, where the mass ux is expected to be the most important, the ow will attain a structural equilibrium rapidly and such an approximation will be adequate. Note that the equality P = " corresponds to the xed point D=Dt < v i >= 0. The algebraic ")=k 8
neglecting <v i >as U i = V i + <v i >. It is clear that this approximation is only valid when <v i >and its gradients are negligible. The data of Dinavahi et al. (1993b) indicates that this approximation is a poor one in the wall bounded ow. In fact, in some portions of the turbulent boundary layer, the mass uxes' contribution to the viscous terms is as high as 25%. Figure 2, taken from Ristorcelli et al. (1993), shows the second cross-stream derivatives of the U i ; V i and <v i >. The mass ux terms also contribute to the pressure and viscous work terms in the mean energy equation. It is clear, given the number of times it occurs in the momentevolution equations, that an expression for the mass ux for general compressible turbulent ows of aerodynamic interest. is necessary. Anevolution equation and a model for <v i >are important: 1) to be able to estimate the importance of <v i >in dierent ows, 2) to know what to do about it when it is important, and 3) to be able to relate experimental values to computational results. 3. An evolution equation for the Favre velocity perturbation Consider the evolution equations for the total velocity and density elds: ; t +( u p ); p = 0 ( u i ); t +( u p u i ); p+2 ikp k u p = p i + f i + ij;j where ij = [u i ; j +u j ; i 2=3u q ; q ij ]. To avery good approximation the viscosity is independent of density: it will be taken to be equal to its local mean value and correlations between the viscosity and velocity will be considered as higher order eects and neglected. The evolution equation for the uctuations around the Favre-mean momentum are obtained by subtracting the evolution equation for the mean momentum <>V i from the equation for u i to obtain an equation for v i.as < v i >= 0, a straightforward time-average of this equation does not produce any results. The equation is rewritten in its nonconservative instantaneous form: v i ; t + (V p + v p )v i ; p +( v p V i ); p < v i v p >; p +2 ikp k v p = p i + f i + ij;j u L vf V i + V i ( v p ); p where u ij =< >[u i ; j +u j ; i 2=3u q ; q ij ] and L vf V i = V i ; t +V p V i ; p +2 ikp k V p F i. The term L vf V i reects the coupling between the uctuating density and the mean ow. Dividing by the total density,, expanding using the binomial theorem, and averaging produces an evolution equation for < v i =< 2 >=<> 2 is the normalized density variance. Keeping only lowest order terms produces > in which successive terms scale as p, where <v i >; t + V p <v i >; p +2 ikp k <v p >= <v p >V i ; p +fv i v p g<>; p <> 1 +<v i v p ; p > +[fv p v i g <v p v i >]; p <f i >+O( p ) 7
The second moment equations for a compressible ow are written, without approximation and after some manipulation, as D=Dt (< >fv i v j g)= <>fv i v p gv j ; p <>fv j v p gv i ; p + Q ij +2=3 <pv k ; k > ij [< pv i > pj + <pv j > ip + <>fv i v j v p g <v j u ip > <v i u jp >]; p + <v j >[ P; i + ik ; k + < ik >; k ]+ <v i >[ P; j + jk ; k + < jk >; k ] <u j ; p u ip > <u i; p u jp > where the mean momentumequations have been used and ij =<>[v i ; j +v j ; i 2=3v q ; q ij ], ij =<>[V i ; j +V j ; i 2=3V q ; q ij ] and u ij =< >[u i ; j +u j ; i 2=3u q ; q ij ]. The form of the equations above reects the following manipulations: 1) The deviatoric part of the pressure-strain correlation is dened as Q ij =< p(v i ; j +v j ; i )> 2=3<pv k ; k > ij and 2) the identity v i = u i + < v i >has been used to rewrite the transport terms in v i variables while keeping the dissipation terms in u i variables. In the equations for the Favreaveraged Reynolds stress the terms arising from surface forces appear naturally in (U i ;u i ) variables while the problem is posed in (V i ;v i )variables. In recasting the Reynolds variables terms in Favre variables the mass ux, < v i >, makes several dierent contributions to the Reynolds stress equations and, of course, to the k =1=2fv j v j gequation. It multiplies the mean ow acceleration which is a new turbulence production mechanism important in ows with strong mean pressure gradients, shocks and expansion fans, and in any ows that have strong streamwise accelerations. The mass ux also contributes to the viscous diusion of the Reynolds stresses a term that is important in the near wall region which is also where the mass ux terms are important. Note that u ij = ij < ij > allows the viscous transport terms to be recast in the Favre variables and that mass ux terms and their derivatives will appear. The mass ux also contributes to the Reynolds stress equations through the pressure ux to which it is coupled by the equation of state: for an ideal gas <pv j >=P[<v j ><> 1 +fv j g T 1 ]. In the adiabatic case the pressure ux can be written, to rst order, as <pv j >=P < v j >< > 1 = c 2 <v j >. Results from some numerical simulations have shown that the pressure and density uctuations of the turbulence passing through a weak shock can be related through such a rule, Lee (1992). This is not found to be true for the wall bounded ow of Dinavahi and Pruett (1993) as shown in Dinavahi et al. (1993). Lee has also found that the pressure ux (as well as the pressure-dilatation) is primarily responsible for the rapid evolution of turbulent kinetic energy downstream of a weak shock. In the mean momentum and mean energy equations the viscous terms appear in Reynolds variables, ij (U) =<>[U i ; j +U j ; i 2=3U q ; q ij ]. When the problem is recast in Favre variables, the viscous terms become functions of the Favre mean velocity and the Favre uctuation mean. It is typical to approximate U i ' V i to close the equation. This involves 6
on the turbulence. b r ij =< u i u j >=<u p u p > An anisotropy tensor based on the Reynolds variables is dened as can also be dened: b f ij =< v i v j >=<v p v p > 1=3 ij. A similar anisotropy tensor using the Favre variables 1=3 ij. An energy weighted deviation of the anisotropy tensor from its density-weighted equivalent is given solely in terms of the Favre uctuation mean: <v p v p >b f ij <u p u p >b r ij =< v i >< v j > 1=3 <v q >< v q > ij : Note that there are only three independent quantities <v i >. >From a heuristic point of view this is pleasantly consistent: the eect of mean ow gradients, V i ; j, is parameterized by the six components of the anisotropy tensor while the eect of the mean density gradients, the vector <>; i, is parameterized by the three components of the mass ux. There are some interesting properties of the mass ux that can be surmised from the above relationships. The most striking, and this is a rigorous result, is that in 1) an isotropic turbulence and or in 2) a statistically stationary homogeneous turbulence with mean velocity gradients and with no mean density gradients, < v i variables are equivalent: U i = V i u i = v i <v i v j >= <u i u j > b f ij = b r ij : >0 and the Reynolds and Favre Similar results hold for relationships between the various moments of u i and v i as can be easily derived. These results come from the following two facts: 1) in an isotropic eld all vector statistics are zero; 2) in a statistically stationary homogeneous eld, whose directional characteristics are solely determined by the mean velocity gradients, which are invariant to coordinate reection, all quantities not invariant to coordinate reection are zero. has been recognized, in the context of compressible turbulence, by Blaisdell (1991). short, in isotropic or homogeneous turbulence without mean density gradients, there is no dierence between the problem posed in Reynolds or Favre variables. This is an important and serious issue aecting the validity of conclusions about the performance of compressible turbulence models which have been developed and tested in homogeneous or isotropic ows. On the other hand it suggests the appropriateness of the incompressible turbulence modeling framework in building models for the compressible ow as they are consistent in the isotropic and homogeneous limit, for arbitrary turbulent Mach number. This In 5
decompositions of the velocity eld. They are related by U i = V i + <v i > u i = v i <v i >: The uctuating Favre mean quanties the dierence between the Favre-mean and the Reynoldsmean velocities, V i and U i,aswell as the dierence between the instantaneous uctuating portions of these two elds. Note that because of the denition of the Favre-average of the Favre-deviation, < v i ><>fv i g=0, <><v i >=<v i >; the time average of the uctuating Favre velocity and the mass ux are equivalent quantities (apart from a scaling by the local mean density). Because of the peculiarities of the densityweighted averaging operation, a second-order statistic, <v i >, can be expressed as the product of two rst-order statistics, < >< v i >. The two phrases mass ux and Favre uctuation mean will be used interchangeably. The primes on the uctuating density have been dropped. As U i = V i + <v i >, the <v i >quantify the dierence between the unweighted or Reynolds mean, U i, and the density-weighted mean, V i, and represent the eects of compressibility through variations in density. Data from Ma = 4:5 DNS computations of Dinavahi and Pruett (1993) in unidirectional developing wall bounded ow indicate that the approximation of U i ' V i in the wall region is inadequate. In this ow, in which M t ' 0:3 and there is a four-fold variation of the mean density over the boundary layer. In data taken from that simulation, shown in Figure 1, it was unexpectedly found that <v 2 >is larger than either U 2 and V 2. It is large enough to cause U 2 and V 2 to have dierent signs. This is an indication that the net uid particle transport and the net momentum transport are in opposite directions. The point is that this is a nominally simple ow, in comparison to those of practical interest, no inection points, no change of geometry, no substantial heat transfer, no cold wall boundary conditions with the concomitant change in sign of the mean density gradient, in which the approximation U 2 ' V 2 was expected to be adequate and they are not even of the same sign. In comparing experimental data and computational results the mass ux plays a role in relating the Reynolds stresses in Favre, v i,variables and Reynolds, u i,variables: <v i v j >= <u i u j >+<v i >< v j >: The moments involving u i are experimentally measured while those involving v i come from the calculations. As it is a vector it describes the anisotropic eects compressibility has 4
no unknown correlations with viscous and pressure terms appear. The evolution equation is simple enough to carry as an additional dierential equation in turbulence simulations, as Zeman and Cole (1991) have proposed. Nonetheless an algebraic truncation of the equation is derived as a further simplication of the problem applicable to most compressible ows of engineering interest. The algebraic truncation, similar to that used in algebraic stress models, assumes a structural equilibrium which relates the material derivative in the uctuating Favre mean equation to the production and dissipation in the kinetic energy equation. The truncation produces a set of three coupled algebraic equations, of the form A ij <v j >=b i. Application of the Cayley-Hamilton theorem produces an explicit closed form expression for the <v i >. The <v i >are found to be proportional to the density gradients with an eddy-viscosity tensor dependent on the Reynolds stress and the mean deformation. The uctuating Favre mean is then related back to mass ux using the well known relation between the two quantities. This article is organized in the following manner. After motivating the investigation in section two, section three describes the derivation of an evolution equation for the mass ux. In the following section an algebraic model for the mass ux is obtained. The general expression for the <v i >is then specialized to several simple mean ows in order to highlight the physics. It is found that, in the limit of isotropic turbulence with negligible mean velocity gradients, the derived expression reduces to the usual scalar eddy-viscosity form derived using a gradient transfer assumption. The model is tested in the Ma =4:5wall bounded DNS of Dinavahi and Pruett (1992). 2. Preliminary exposition In general, upper case letters will be used to denote mean quantities except in the case of the mean density, <>, since which has no convenient upper case form. The averaging operation is indicated using the angle brackets for time means, < v i v j >, and the curly brackets for the density-weighted or Favre mean, fv i v j g, where <>fv i v j g=< v i v j > and the asterisk denotes the full eld, =< > + 0. The dependent variables are decomposed according to u i = U i + u i where <u i >=0 u = i V i+v i where fv i g =0 = <>+ 0 where < 0 >=0 p = P+p where <p>=0 T = T+ where fg =0 As both the Reynolds and the Favre velocities appear naturally in the evolution equations for a compressible turbulence it is necessary to carry both the Favre and the Reynolds 3
Others have recognized the importance of the mass ux and several models have been proposed. Taulbee and VanOsdol (1991) have derived a modeled equation for the mass ux. In their equation they keep the correlations with the surface forces which are modeled assuming a homogeneous turbulence and the validity of Morkovin's hypothesis. In the present asymptotic derivation these terms scale with the density intensity and are found to be of higher order; the present asymptotic derivation of the transport equation for the mass ux keeps only the zeroeth terms to keep the model simple and to avoid the loss of accuracy associated with the individual approximations made in many models. They also use a gradient transfer assumption for the turbulent diusion. Due to the dierent manipulation to obtain an equation for the the Favre uctuation equation, the turbulent diusion is found to be proportional to the dierence between the Favre and Reynolds averaged Reynolds stresses; as this is a higher order term, scaling as the density intensity, there is no need to model it. The Taulbee and VanOsdol model requires the solution of two modeled dierential equations, one for the mass ux and one for the density variance which appears as a source term in their modeled mass ux equation. In the present derivation of the evolution equation for the mass ux only one unknown, the covariance with the uctuating dilatation, requires modeling. There is no need for a separate equation for the density variance. Zeman and Coleman (1991) have also proposed a mass ux model. Their modeled equation, which has been tested in the turbulence through a shock simulations of Lee (1992) is very similar to the one derived in this article. They also propose an algebraic expression for the mass ux to which the present model simplies to, in the limit of negligible mean deformation. Our work has shown that the inclusion of the mean velocity gradients is essential to capturing the near wall maxima of the mass ux. After all, the mean velocity gradients are a major portion of the production terms of the mass ux in its transport equation. Rubesin (1990) has also proposed a mass ux model. It assumes that 1) the uctuations obey a polytropic gas law 2) the specic heats are constant allowing the uctuating density to be written in terms of the uctuating enthalpy and that 3) the uctuating enthalpy can be related to the mean enthalpy using a gradient transfer hypothesis. The Rubesin model requires the polytropic index as an input. Dinavahi et al. (1993) and Ristorcelli et al. (1993) in a temporal DNS have shown that the polytropic index varies substantially over the width of the turbulent boundary layer. The Rubesin model also predicts a mass ux only when there is a heat ux, while the present model derived from the exact evolution equation, predicts a mass ux whenever there are mean density gradients. The present model for the mass ux starts with the exact evolution equation for the mass ux. An equation for the uctuating Favre mean is then developed in a power series in the uctuating density intensity. To zeroeth order there is only one unknown correlation and 2
1. Introduction This article presents a derivation of a representation for the non-zero rst-moment of the uctuating velocity eld, the time average of the uctuating component of the Favre velocity, <v i >. Mathematically <v i >represents the dierence between unweighted and density weighted averages of the velocity eld and is therefore a measure of the eects of compressibility through variations in density. It plays an important role in parameterizing the anisotropic eects of compressibility associated with the mean dilatation and gradients in the mean velocity and density. Experimentally it is an important quantity that allows Favreaveraged numerical results to be related to time-averaged experimental results. The need to consider this quantity is motivated by its frequent contributions to the rst and second-order moment equations in two-equation k " type turbulence closures as well as in Reynolds stress closures. In the mean momentum equations the mass ux makes a contribution to the viscous terms. In the mean energy equations the mass ux makes a contribution to the viscous, the pressure work, and the pressure ux terms. In the Reynolds stress equations the viscous terms appear naturally in Reynolds variables while the problem is posed in Favre variables. In the process of splitting the viscous terms into the viscous transport terms, carried in Favre variables, and the dissipation terms, carried in Reynolds variables, important contributions from the mass ux appear. The accurate accounting of these terms is important for any consistent near wall modeling and the retention of the mass ux terms is important in complex compressible turbulent ows. These contributions have been investigated in detail in Ristorcelli (1993). The mass ux also determines the importance of two production mechanisms one due to the acceleration of the mean ow and the other due to viscous eects associated with the Favre uctuation mean. Many of these contributions are neglected in turbulence closure models. This is a result of assuming that the Favre mean velocities are suitable approximations to the Reynolds mean velocities. This approximation is not appropriate in complex ows of practical interest. The retention of the mass ux terms will be necessary in complex compressible turbulent ows: these include ows in which there are mean density gradients due to large Mach number, combustion, separation or reattachment (inection points), cold wall boundary conditions, mean dilatation, shocks, adverse pressure gradients, or strong streamwise accelerations. Even in this nominally simple compressible ow, such as a supersonic wall bounded boundary layer which has a four-fold variation of the mean density over the width of the boundary layer, the mass ux is not negligible. Dinavahi et al. (1993b), in a Mach 4.5 wall bounded DNS, has found that the cross-stream Favre mean and Reynolds mean velocities have dierent signs attesting to the fact that the mass ux is not small with respect to the mean velocities. 1
A REPRESENTATION FOR THE TURBULENT MASS FLUX CONTRIBUTION TO REYNOLDS-STRESS AND TWO-EQUATION CLOSURES FOR COMPRESSIBLE TURBULENCE J.R. Ristorcelli 1 Institute for Computer Applications in Science and Engineering NASA Langley Research Center Hampton, VA, 23681 ABSTRACT The turbulent mass ux, or equivalently the uctuating Favre velocity mean, appears in the rst and second moment equations of compressible k " and Reynolds stress closures. Mathematically it is the dierence between the unweighted and density-weighted averages of the velocity eld and is therefore a measure of the eects of compressibility through variations in density. It appears to be fundamental to an inhomogeneous compressible turbulence, in which itcharacterizes the eects of the mean density gradients, in the same way the anisotropy tensor characterizes the eects of the mean velocity gradients. An evolution equation for the turbulent mass ux is derived. A truncation of this equation produces an algebraic expression for the mass ux. The mass ux is found to be proportional to the mean density gradients with a tensor eddy-viscosity that depends on both the mean deformation and the Reynolds stresses. The model is tested in a wall bounded DNS at Mach 4.5 with notable results. 1 This research was supported by the National Aeronautics and Space Administration under NASA Contract No. NAS1-19480 while the author was in residence at the Institute for Computer Applications in Science and Engineering (ICASE), NASA Langley Research Center, Hampton, VA 23681. i